Time Fractional Convection Diffusion Equation Research Articles

Fractional diffusion equations interpolate between damping and waves

Abstract The behaviour of the solutions of the time-fractional diffusion equation,
based on the Caputo derivative, is studied and its dependence on the fractional
exponent is analysed. The time-fractional convection-diffusion equation is also solved
and an application to Pennes bioheat model is presented. Generically, a wave-like
transport at short times passes over to a diffusion-like behaviour at later times.

Singularly perturbed time-fractional convection–diffusion equations via exponential fitted operator scheme

In this paper, we proposed an accurate ϵ-uniformly convergent numerical method to solve singularly perturbed time-fractional convection–diffusion equations via exponential fitted operator scheme. The time-fractional derivative is defined in the sense of Caputo with order η∈(0,1). The time-fractional derivative is discretized by employing the Crank–Nicolson method on a uniform mesh, and an exponential fitted operator scheme along with the standard upwind method is used to mesh-grid the space domain. The truncation error and uniform stability of the discretized problems are examined in order to prove the parameter uniform convergence of the proposed scheme. It is demonstrated that the scheme is ϵ-uniformly convergent of order O((Δt)2−η+Δx), where Δt and Δx represent the step sizes of the time and space domains, respectively. Two numerical examples are provided in order to assess the accuracy of the suggested scheme and validate the theoretical concepts discussed. To demonstrate the efficiency of the numerical scheme presented, comparisons have been made with the numerical solution obtained by the finite difference method that exists in the literature. Consequently, it is observed that the results obtained by the present scheme are more accurate and have a better convergence rate.

Error analysis of a high-order fully discrete method for two-dimensional time-fractional convection-diffusion equations exhibiting weak initial singularity

Error analysis of a high-order fully discrete method for two-dimensional time-fractional convection-diffusion equations exhibiting weak initial singularity

Efficient Numerical Solutions for Fuzzy Time Fractional Convection Diffusion Equations Using Two Explicit Finite Difference Methods

In this study, we explore fractional partial differential equations as a more generalized version of classical partial differential equations. These fractional equations have shown promise in providing improved descriptions of certain phenomena under specific circumstances. The main focus of this paper comprises the development, analysis, and application of two explicit finite difference schemes to solve an initial boundary value problem involving a fuzzy time fractional convection–diffusion equation with a fractional order in the range of 0≤ ξ ≤ 1. The uniqueness of this problem lies in its consideration of fuzziness within both the initial and boundary conditions. To handle the uncertainty, we propose a computational mechanism based on the double parametric form of fuzzy numbers, effectively converting the problem from an uncertain format to a crisp one. To assess the stability of our proposed schemes, we employ the von Neumann method and find that they demonstrate unconditional stability. To illustrate the feasibility and practicality of our approach, we apply the developed scheme to a specific example.

A space-time Galerkin Müntz spectral method for the time fractional Fokker–Planck equation

In this paper, we propose a space-time Galerkin spectral method for the time fractional Fokker–Planck equation. This approach is based on combining temporal Müntz Jacobi polynomials spectral method with spatial Legendre polynomials spectral method. Based on the well-posedness and regularity for the re-scaled problem of a linear model problem which reflects the main difficulty for solving the equivalent equation (i.e. the time fractional convection-diffusion equation): the singularity of the solution in time, we explain in detail why we use the Müntz polynomials to approximate in time. The well-posedness and stability of the discrete scheme as well as its continuous problem are established. Moreover, the error estimation of the space-time approach is derived. We find that the proposed method can attain spectral accuracy regardless of whether the solution of the original equation is smooth or non-smooth. Numerical experiments substantiate the theoretical results.

Modeling anomalous transport in fractal porous media: A study of fractional diffusion PDEs using numerical method

Abstract Fractional diffusion partial differential equation (PDE) models are used to describe anomalous transport phenomena in fractal porous media, where traditional diffusion models may not be applicable due to the presence of long-range dependencies and non-local behaviors. This study presents an efficient hybrid meshless method to the compute numerical solution of a two-dimensional multiterm time-fractional convection-diffusion equation. The proposed meshless method employs multiquadric-cubic radial basis functions for the spatial derivatives, and the Liouville-Caputo derivative technique is used for the time derivative portion of the model equation. The accuracy of the method is evaluated using error norms, and a comparison is made with the exact solution. The numerical results demonstrate that the suggested approach achieves better accuracy and computationally efficient performance.

Uniformly convergent numerical method for time-fractional convection–diffusion equation with variable coefficients

This paper presents a uniformly convergent numerical scheme for singularly perturbed fractional order convection–diffusion equations with variable coefficients. First, the time-fractional derivative is considered in the Caputo sense and treated using the implicit Euler method. Then, a uniformly convergent numerical scheme based on the nonstandard finite difference method is developed. The technique is proved rigorously for parameter-uniform convergence. By numerical experimentation, it is also validated that the computational results agree with theoretical results.

Fibonacci wavelet method for time fractional convection–diffusion equations

This study concentrates on time fractional convection–diffusion equations (TFCDEs) with variable coefficients and their numerical solutions. Caputo derivative is used to calculate the time fractional order derivatives. In order to give an approximate solution to the TFCDE, an effective approach is proposed utilizing Fibonacci wavelet and block pulse functions. The Fibonacci wavelets operational matrices of fractional order integration are constructed. By combining the collocation technique, they are used to simplify the fractional model to a collection of algebraic equations. The suggested approach is quite practical for resolving issues of this nature. The comparison and analysis with other approaches demonstrate the effectiveness and precision of the suggested approach.

Mapping techniques for collocation method of time-fractional convection–diffusion equations in domains with cracks

This paper proposes numerical methods that effectively deal with time-fractional convection–diffusion equations containing crack singularities. To deal with singularities, we design the geometrical mapping whose push-forward from the parameter space into the physical space generates point singularity functions based on the parametrization of the circular arc and NURBS (non-uniform rational B-spline). We adopt the collocation method with B-spline basis functions to approximate the solution in the spatial direction and enrich the approximation space by k-refinements in IGA (Isogeometric Analysis). For the discretization along the temporal direction, we employ the explicit Predictor-Corrector (PC) scheme that has the order 2−ν and 3−ν of the truncation error for the linear and quadratic interpolation, respectively. Taking advantage of the NURBS geometrical mapping, we demonstrate the performance of the proposed methods applying to time-fractional convection–diffusion equations with nonlinear terms on curved domains containing crack singularities.

Computational analysis of time-fractional models in energy infrastructure applications

In this paper, we propose an effective numerical method to solve the one- and two-dimensional time-fractional convection-diffusion equations based on the Caputo derivative. The presented approach employs a hybrid method that combines Lucas and Fibonacci polynomials with the Caputo derivative definition. The main objective is to transform the problem into a time-discrete form utilizing the Caputo derivative technique and then approximate the function's derivative using Fibonacci polynomials. To evaluate the efficiency and accuracy of the proposed technique, we apply it to one- and two-dimensional problems and compare the results with the exact as well as with existing methods in recent literature. The comparison demonstrates that the proposed approach is highly efficient, accurate and ease to implement.

A nonuniform L2-1$_\sigma$/LDG method for the Caputo-Hadamard time-fractional convection-diffusion equation

A nonuniform L2-1$_\sigma$/LDG method for the Caputo-Hadamard time-fractional convection-diffusion equation

Fourth-Order Numerical Solutions for a Fuzzy Time-Fractional Convection–Diffusion Equation under Caputo Generalized Hukuhara Derivative

The fuzzy fractional differential equation explains more complex real-world phenomena than the fractional differential equation does. Therefore, numerous techniques have been timely derived to solve various fractional time-dependent models. In this paper, we develop two compact finite difference schemes and employ the resulting schemes to obtain a certain solution for the fuzzy time-fractional convection–diffusion equation. Then, by making use of the Caputo fractional derivative, we provide new fuzzy analysis relying on the concept of fuzzy numbers. Further, we approximate the time-fractional derivative by using a fuzzy Caputo generalized Hukuhara derivative under the double-parametric form of fuzzy numbers. Furthermore, we introduce new computational techniques, based on fuzzy double-parametric form, to shift the given problem from one fuzzy domain to another crisp domain. Moreover, we discuss some stability and error analysis for the proposed techniques by using the Fourier method. Over and above, we derive several numerical experiments to illustrate reliability and feasibility of our proposed approach. It was found that the fuzzy fourth-order compact implicit scheme produces slightly better results than the fourth-order compact FTCS scheme. Furthermore, the proposed methods were found to be feasible, appropriate, and accurate, as demonstrated by a comparison of analytical and numerical solutions at various fuzzy values.

A high-order numerical scheme based on graded mesh and its analysis for the two-dimensional time-fractional convection-diffusion equation

This work proposes a numerical method for solving the two-dimensional time-fractional convection-diffusion (TFCD) equation. The time-fractional derivative is considered in the sense of Caputo. The considered problem with Caputo temporal fractional derivative generally shows a weak singularity at the initial time. In order to handle the weak singularity at t = 0 , the time-fractional derivative is discretized on a non-uniform graded mesh. A high-order compact operator is employed for approximating the derivatives in space directions. The resulting system of equations is then solved using the Alternating Direction Implicit (ADI) approach. We prove the stability and convergence of this scheme. Numerical experiment is performed to demonstrate the applicability and accuracy of the method. The considered numerical example subjected to nonsmooth solution confirms that the method has an order min ⁡ ( 2 − α , r α , 2 α + 1 ) in the temporal direction, where α ∈ ( 0 , 1 ) is the order of the fractional derivative and r is a parameter used in construction of the graded mesh and has a rate of convergence of order four in spatial direction. The proposed method on graded mesh has an advantage in terms of numerical accuracy over the method on the uniform mesh. The CPU time (in seconds) for the present method is also provided.

A New Modified Analytical Approach for the Solution of Time-Fractional Convection–Diffusion Equations With Variable Coefficients

In this article, a new modification of the Adomian decomposition method is performed for the solution fractional order convection–diffusion equation with variable coefficient and initial–boundary conditions. The solutions of the suggested problems are calculated for both fractional and integer orders of the problems. The series of solutions of the problems with variable coefficients have been provided for the first time. To verify and illustrate our new technique, four numerical examples are presented and solved by using the proposed technique. The derived results are plotted, and the dynamics are shown for both fractional and integer orders of the problems. An excellent variation among the solutions at various fractional orders is observed. It is analyzed that the new technique based on the Adomian decomposition method is accurate and effective. The present method fits both the initial and boundary conditions with double approximations simultaneously, which increases the accuracy of the present method. For the first time, the present technique is used for the solutions of the problems with variable coefficients along with initial and boundary conditions. It is therefore suggested to apply the present procedure for the solutions of other problems with variable order and coefficients along with initial and boundary conditions.

High-order numerical algorithms for the time-fractional convection–diffusion equation

In this paper, efficient methods are derived for seeking a numerical solution to the time-fractional convection–diffusion equation whose solution very likely exhibits a weak regularity at the starting time. Here, the time-fractional derivative in the Caputo sense with order in is discretized by the - methods with uniform and non-uniform meshes and the spatial derivative is approximated by the local discontinuous Galerkin (LDG) finite element methods. The fully discrete schemes for both situations are established and analyzed. It is shown that the derived schemes are numerically stable and convergent. Finally, some numerical examples are performed to testify the effectiveness of the obtained algorithm.

Semi-discretized numerical solution for time fractional convection–diffusion equation by RBF-FD

In this paper, a numerical method based on radial basis functions finite difference (RBF-FD) has been developed for solving the time fractional convection–diffusion equation. We first approximate the time fractional derivative by FD and obtain the semi-discretized scheme. The unconditional stability and convergence of the semi-discretized scheme are given. After that, we use the multiquadric RBF to approximate the spatial derivatives in the numerical experiments. The aim of this paper is to show that the RBF-FD method is better than the classical FD method for solving our mentioned equation. Finally, two numerical examples are proposed respectively to verify the correctness of our theoretical analysis and to demonstrate the superiority of the RBF-FD method.

On the Numerical Approximation of Three-Dimensional Time Fractional Convection-Diffusion Equations

In this paper, we present an efficient method for the numerical investigation of three-dimensional non-integer-order convection-diffusion equation (CDE) based on radial basis functions (RBFs) in localized form and Laplace transform (LT). In our numerical scheme, first we transform the given problem into Laplace space using Laplace transform. Then, the local radial basis function (LRBF) method is employed to approximate the solution of the transformed problem. Finally, we represent the solution as an integral along a smooth curve in the complex left half plane. The integral is then evaluated to high accuracy by a quadrature rule. The Laplace transform is used to avoid the classical time marching procedure. The radial basis functions are important tools for scattered data interpolation and for solving partial differential equations (PDEs) of integer and non-integer order. The LRBF and global radial basis function (GRBF) are used to produce sparse collocation matrices which resolve the issue of the sensitivity of shape parameter and ill conditioning of system matrices and reduce the computational cost. The application of Laplace transformation often leads to the solution in complex plane which cannot be generally inverted. In this work, improved Talbot’s method is utilized which is an efficient method for the numerical inversion of Laplace transform. The stability and convergence of the method are discussed. Two test problems are considered to validate the numerical scheme. The numerical results highlight the efficiency and accuracy of the proposed method.

Numerical methods for time-fractional convection-diffusion problems with high-order accuracy

Abstract In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α \alpha ( 1 < α < 2 1\lt \alpha \lt 2 ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H 1 {H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O ( τ 3 − α + h 1 4 + h 2 4 ) O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}) , where τ \tau is the temporal grid size and h 1 {h}_{1} , h 2 {h}_{2} are spatial grid sizes in the x x and y y directions, respectively. It is proved that the method can even attain ( 1 + α ) \left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.

A fast finite difference/RBF meshless approach for time fractional convection-diffusion equation with non-smooth solution

In this paper, we propose a fast finite difference/RBF meshless method for time fractional convection-diffusion equation with singularity at the initial moment. Firstly, the time fractional derivative term is discretized by classical L1 format on graded meshes, and the spatial derivative term is approximated by meshless method. The meshless method does not need to construct meshes, which is more conducive for dealing with complex regions or complex boundary conditions. In addition, we adopt proper orthogonal decomposition (POD) technology to derive reduced-order RBF meshless extrapolation algorithm, which effectively reduces the amount of calculation and improves the calculation efficiency. Finally, the accuracy and efficiency of the method are verified by numerical examples.

The Chebyshev collocation method for a class of time fractional convection‐diffusion equation with variable coefficients

In this paper, an efficient and accurate computational scheme based on the Chebyshev collocation method and finite difference approximation is proposed to solve the time‐fractional convection‐diffusion equation (TFCDE) on a finite domain. The time fractional‐order derivativeμ ∈ (0, 1]is considered in the Caputo sense. The finite‐difference approximation is used in time direction while the Chebyshev collocation method is used in space direction to reduce the TFCDE into a system of algebraic equations. We also illustrate the error and convergence analysis of the proposed scheme. The proposed method is very convenient for solving such problems since the initial and boundary conditions are automatically taken into account. The efficiency and accuracy of the proposed algorithm are examined through some examples and comparisons with existing methods.